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entropy1
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Why does an Observable have to be Hermitian, and why do the eigenstates and eigenvalues have to respresent the possible measured values? Is is by definition? What is the origin of this convention?
As the discussion here at https://www.physicsforums.com/posts/5513644 shows, observables don't have to be Hermitian and often aren't. Only those to be measured by an ideal von-Neumann measurement must be Hermitian, because this guarantees the existence of the projections required by Born's rule.entropy1 said:a
Introduced by fiat in chapter II of PAM Dirac, The Principles of Quantum Mechanics.entropy1 said:Why does an Observable have to be Hermitian, and why do the eigenstates and eigenvalues have to respresent the possible measured values? Is is by definition? What is the origin of this convention?
Only Hermitian operators are guaranteed to have only real eigenvalues... and it seems a reasonable enough postulate that any measured value must be a real number.entropy1 said:Why does an Observable have to be Hermitian?
That's another postulate. We use these postulates because they work; that is, the mathematics that follows from them makes accurate quantitative predictions about how the universe behaves. You might find wikipedia's article on the history of the mathematical formalism interesting: https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#History_of_the_formalismand why do the eigenstates and eigenvalues have to represent the possible measured values? Is is by definition? What is the origin of this convention?
The frequency of a damped harmonic oscillator is a measurable, complex number. It occurs over and over in the electrodynamics of circuits.Nugatory said:any measured value must be a real number.
MichPod1 said:One may start with a postulate that for a particular observable the states for which it has uniquly defined measured value ##a_i## are orthogonal: ##\langle\psi_i|\psi_j\rangle=0## for ##i\neq j##
Without loosing of generality, we can also normalize them and further consider them orthonormal. ##\langle\psi_i|\psi_j\rangle=\delta_{ij}##
Yes, and allowing in place of the rank 1 projectors arbitrary positive definite Hermitian operators leads in this way naturally to POVMs. This shows that the latter are far more natural. The emphasis on the special case of rank 1 projectors (leading to Born's rule) is a historical accident only.Truecrimson said:So arranging eigenvectors and eigenvalues into a Hermitian operator can be thought of as merely a convenient way to calculate an average.
In general, collapse is not to an eigenstate since observations are almost never perfect. Thus this gives no argument in favor of Hermitian operators.entropy1 said:I was thinking, do we use Hermitian Operators because they have eigenvectors, and eigenvectors are compatible with the phenomenon of collapse that we observe?
No. Operators don't need to be Hermitian to have eigenvectors.entropy1 said:I was thinking, do we use Hermitian Operators because they have eigenvectors
No, because collapse is not a phenomenon that we observe.and eigenvectors are compatible with the phenomenon of collapse that we observe?
I learned that when you measure, for instance, electron spin, getting as a result (for instance) 'up spin' has as a consequence that subsequent measurements along the same axis gives a 100% probability of yielding the spin along that same axis. Are you saying that we haven't observed collapsing the state of the spin there?Nugatory said:because collapse is not a phenomenon that we observe.
Yes, this follows from the mathetatical formalism of quantum mechanics. But...entropy1 said:I learned that when you measure, for instance, electron spin, getting as a result (for instance) 'up spin' has as a consequence that subsequent measurements along the same axis gives a 100% probability of yielding the spin along that same axis.
Yes, that is what I and other people other people have been saying, for more posts now than I care to count.Are you saying that we haven't observed collapsing the state of the spin here?
How would you do this? Single particle states are very fragile.entropy1 said:when you measure, for instance, electron spin, getting as a result (for instance) 'up spin' has as a consequence that subsequent measurements along the same axis
A. Neumaier said:The frequency of a damped harmonic oscillator is a measurable, complex number. It occurs over and over in the electrodynamics of circuits.
A. Neumaier said:The frequency of a damped harmonic oscillator is a measurable, complex number. It occurs over and over in the electrodynamics of circuits.
Derek Potter said:It's not measurable as a complex number though.
What we actually have is a typically a second order linear circuit modeled by a couple of linear equations. The general solutions are then complex exponentials. But physics constrains the actual solutions to occur in superposition - the familiar expansion of a sine or cosine as the sum of two imaginary exponentials.
So whilst you can correctly say the natural frequency is complex, the physical observables still have to be real.
Traruh Synred said:A complex number is just two real numbers with a rule for 'multiplying' them. A simple rate is just one real number. If it varies in time or space it takes more than one real number to describe the dependence. The measurements of dependence are just as real as anything. There's in fact even then no need to use 'complex' or 'imaginary' numbers, but that notation gives a more elegant formulation. It's not different than tensors or matrices in other context. Just notation, but good notation.
A damped harmonic oscillator has a physically meaningful complex frequency.Derek Potter said:l. A. Neumaier gave complex frequency as a counter-example. For it to be a valid counter-example it would have to be complex-valued, which it is, and physically possible, which, in general, complex frequencies are not.
A. Neumaier said:A damped harmonic oscillator has a physically meaningful complex frequency.
Derek Potter said:Yes I know what a complex number is.
But we are not talking about whether complex numbers are handy. We are talking about physical properties necessarily being real. A. Neumaier gave complex frequency as a counter-example. For it to be a valid counter-example it would have to be complex-valued, which it is, and physically possible, which, in general, complex frequencies are not. That is to say, physically possible solutions are a subspace of the complex solution space. (They are confined to solutions that are real in the time domain.) Do you see what I'm saying?
You can't have it both ways. If you want to say that a complex frequency is actually a real-valued function then it isn't an example of a complex-valued observable.Traruh Synred said:An 'imaginary' frequency is just a plain old exponential or a 'sinh' or something. Nothing unreal about it. It's just words. A frequency could be called an 'imaginary' value of a a decay or growth rate. All the numbers whether in pairs or not are, of course, numbers.
Traruh Synred said:Sure I can have it both ways. It's just words. Complex observable is just a word for needing two numbers to describe the measurement. It get's elegant when the multiplication is defined to form a 'ring' . I can _call _that measuring a complex variable. In the other sense all the numbers that get measured a 'real' whether call them 'imaginary part' or not. For a rate at a single time you only need one number to specify it.
In a damped harmonic oscillator, the complex-valued frequency has a nonzero real and imaginary part. The real observables are the points on the oscillating curve; the observable complex frequency is extracted from these and produces the physical way of summarizing the behavior of the oscillator.Derek Potter said:you represent a real-valued observable by a complex number.
A. Neumaier said:In a damped harmonic oscillator, the complex-valued frequency has a nonzero real and imaginary part.
Derek Potter said:Well, I'll try one more time.
A. Neumaier said:In a damped harmonic oscillator, the complex-valued frequency has a nonzero real and imaginary part.
Nice post-editing :)A. Neumaier said:The real observables are the points on the oscillating curve; the observable complex frequency is extracted from these and produces the physical way of summarizing the behavior of the oscillator.
It is always the summary that carries the physics. Without summarizing what happens in Nature we cannot form a single concept. Every observable is an abstraction of the real thing, and as an abstraction it may be a real number, a complex number, or an even more complicated object such as a vector or a tensor.
Take a wire and bend it to a sine wave :)Derek Potter said:I am surprised that nobody else seems to be interested in whether the idea of complex observables makes sense physically.
Observables are physical quantities that can be measured and observed in a system. They can include properties such as position, velocity, mass, temperature, and more.
In quantum mechanics, observables are represented by operators, which act on the wave function of a system to determine the possible values that can be observed for that observable. The measurement of an observable in quantum mechanics can give a probabilistic result rather than a definite value.
In classical physics, observables are properties of a system that can be directly measured and observed without affecting the system. They are described by classical mechanics and follow deterministic laws.
The uncertainty principle in quantum mechanics states that it is impossible to know the exact values of certain pairs of observables, such as position and momentum, simultaneously. This is because the act of measuring one observable affects the other, leading to inherent uncertainty in the measurements.
Yes, there are limitations to what can be considered an observable. In quantum mechanics, only certain quantities that can be measured and observed are considered observables. Additionally, some physical quantities may be impossible to measure due to technological limitations or the principles of quantum mechanics.